MULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS

Transcription

1 ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS CAILLO DE LELLIS AND EANUELE SPADARO Abstract. We prove several results on Almgren s multple valued functons and ther lnks to ntegral currents. In partcular, we gve a smple proof of the fact that a Lpschtz multple valued map naturally defnes an nteger rectfable current; we derve explct formulae for the boundary, the mass and the frst varatons along certan specfc vectorfelds; and explot ths connecton to derve a delcate reparametrzaton property for multple valued functons. These results play a crucal role n our new proof of the partal regularty of area mnmzng currents [5, 6, 7]. 0. Introducton It s known snce the poneerng work of Federer and Flemng [10] that one can naturally assocate an nteger rectfable current to the graph of a Lpschtz functon n the Eucldean space, ntegratng forms over the correspondng submanfold, endowed wth ts natural orentaton. It s then possble to derve formulae for the boundary of the current, ts mass and ts frst varatons along smooth vector-felds. oreover, all these formulae have mportant Taylor expansons when the current s suffcently flat. In ths paper we provde elementary proofs for the correspondng facts n the case of Almgren s multple valued functons (see [4] for the relevant defntons). The connecton between multple valued functons and ntegral currents s crucal n the analyss of the regularty of area mnmzng currents for two reasons. On the one hand, t provdes the necessary tools for the approxmaton of currents wth graphs of multple valued functon. Ths s a fundamental dea for the study of the regularty of mnmzng currents n the classcal sngle-vaued case, and t also plays a fundamental role n the proof of Almgren s partal regularty result (cf. [1, 5]). In ths perspectve, explct expressons for the mass and the frst varatons are necessary to derve the rght estmates on the man geometrc quanttes nvolved n the regularty theory (cf. [5, 6, 7]). On the other hand, the connecton can be exploted to nfer nterestng conclusons about the multple valued functons themselves. Ths pont of vew has been taken frutfully n many problems for the case of classcal functons (see, for nstance, [11, 12] and the references theren), and has been recently exploted n the multple valued settng n [3, 14]. The prototypcal example of nterest here s the followng: let f : R m Ω R n be a Lpschtz map and Gr(f) ts graph. If the Lpschtz constant of f s small and we change coordnates n R m+n wth an orthogonal transformaton close to the dentty, then the set Gr(f) s the graph of a Lpschtz functon f over some doman Ω also n the new system of coordnates. In fact t s easy to see that there exst sutable maps Ψ and Φ such that f(x) = Ψ ( x, f(φ(x)) ). In the multple valued 1

2 2 CAILLO DE LELLIS AND EANUELE SPADARO case, t remans stll true that Gr(f) s the graph of a new Lpschtz map f n the new system of coordnates, but we are not aware of any elementary proof of such statement, whch has to be much more subtle because smple relatons as the one above cannot hold. It turns out that the structure of Gr(f) as ntegral current gves a smple approach to ths and smlar ssues. Several natural estmates can then be proved for f, although more nvolved and much harder. The last secton of the paper s dedcated to these questons; more careful estmates obtaned n the same ven wll also be gven n [6], where they play a crucal role. ost of the conclusons of ths paper are already establshed, or have a counterpart, n Almgren s monograph [1], but we are not always able to pont out precse references to statements theren. However, also when ths s possble, our proofs have an ndependent nterest and are n our opnon much smpler. ore precsely, the materal of Sectons 1 and 2 s covered by [1, Sectons ], where Almgren deals wth general flat chans. Ths s more than what s needed n [5, 6, 7], and for ths reason we have chosen to treat only the case of nteger rectfable currents. Our approach s anyway smpler and, nstead of relyng, as Almgren does, on the ntersecton theory of flat chans, we use rather elementary tools. For the theorems of Secton 3 we cannot pont out precse references, but Taylor expansons for the area functonal are ubqutous n [1, Chapters 3 and 4]. The theorems of Secton 4 do not appear n [1], as Almgren seems to consder only some partcular classes of deformatons (the squeeze and squash, see [1, Chapter 5]), whle we derve farly general formulas. Fnally, t s very lkely that the conclusons of Secton 5 appear n some form n the constructon of the center manfold of [1, Chapter 4], but we cannot follow the ntrcate arguments and notaton of that chapter. In any case, our approach to reparametrzons of multple valued maps seems more flexble and powerful, capable of further applcatons, because, as t was frst realzed n [4], we can use tools from metrc analyss and metrc geometry developed n the last 20 years. Acknowledgments The research of Camllo De Lells has been supported by the ERC grant agreement RA (Regularty for Area nmzng currents), ERC The authors are warmly thankful to Bll Allard for several enlghtenng conversatons and hs constant enthusastc encouragement; and very grateful to Luca Spolaor and atteo Focard for carefully readng a prelmnary verson of the paper and for ther very useful comments. Camllo De Lells s also very thankful to the Unversty of Prnceton, where he has spent most of hs sabbatcal completng ths and the papers [5, 6, 7]. 1. Q-valued push-forwards We use the notaton, for: the eucldean scalar product, the naturally nduced nner products on p-vectors and p-covectors and the dualty parng of p-vectors and p-covectors; we nstead restrct the use of the symbol to matrx products. Gven a C 1 m-dmensonal submanfold Σ R N, a functon f : Σ R k and a vector feld X tangent to Σ, we denote by D X f the dervatve of f along X, that s D X f(p) = (f γ) (0) whenever γ s a smooth curve on Σ wth γ(0) = p and γ (0) = X(p). When k = 1, we denote by f the vector feld tangent to Σ such that f, X = D X f for every tangent vector feld X. For general

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